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Simplifying 14z4 + -42z3 + 21z2 = 0 Reorder the terms: 21z2 + -42z3 + 14z4 = 0 Solving 21z2 + -42z3 + 14z4 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '7z2'. 7z2(3 + -6z + 2z2) = 0 Ignore the factor 7.Subproblem 1
Set the factor 'z2' equal to zero and attempt to solve: Simplifying z2 = 0 Solving z2 = 0 Move all terms containing z to the left, all other terms to the right. Simplifying z2 = 0 Take the square root of each side: z = {0}Subproblem 2
Set the factor '(3 + -6z + 2z2)' equal to zero and attempt to solve: Simplifying 3 + -6z + 2z2 = 0 Solving 3 + -6z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -3z + z2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3z + -1.5 + z2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3z + z2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3z + z2 = 0 + -1.5 -3z + z2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3z + z2 = -1.5 The z term is -3z. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3z + 2.25 + z2 = -1.5 + 2.25 Reorder the terms: 2.25 + -3z + z2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + -3z + z2 = 0.75 Factor a perfect square on the left side: (z + -1.5)(z + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (z + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
z + -1.5 = 0.866025404 Simplifying z + -1.5 = 0.866025404 Reorder the terms: -1.5 + z = 0.866025404 Solving -1.5 + z = 0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 0.866025404 + 1.5 z = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 z = 2.366025404 Simplifying z = 2.366025404Subproblem 2
z + -1.5 = -0.866025404 Simplifying z + -1.5 = -0.866025404 Reorder the terms: -1.5 + z = -0.866025404 Solving -1.5 + z = -0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -0.866025404 + 1.5 z = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 z = 0.633974596 Simplifying z = 0.633974596Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.366025404, 0.633974596}Solution
z = {0, 2.366025404, 0.633974596}
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